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Ch. 3 - Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 3 - DerivativesProblem 21a
Chapter 3, Problem 21a

Economics


Marginal cost Suppose that the dollar cost of producing x washing machines is c(x) = 2000 + 100x − 0.1x².


a. Find the average cost per machine of producing the first 100 washing machines.

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1
First, understand the given cost function: c(x) = 2000 + 100x - 0.1x², where c(x) represents the total cost of producing x washing machines.
To find the average cost per machine for the first 100 machines, we need to calculate the total cost for producing 100 machines, which is c(100).
Substitute x = 100 into the cost function: c(100) = 2000 + 100(100) - 0.1(100)².
Calculate c(100) to find the total cost of producing 100 machines.
Finally, divide the total cost c(100) by 100 to find the average cost per machine: Average cost = c(100) / 100.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Average Cost

Average cost is calculated by dividing the total cost by the number of units produced. It provides insight into the cost efficiency of production. For the given function c(x), the average cost per machine for producing x washing machines is c(x)/x, which helps determine the cost per unit for a specific production level.
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Average Value of a Function

Cost Function

A cost function, such as c(x) = 2000 + 100x − 0.1x², represents the total cost of producing x units. It includes fixed costs, variable costs, and sometimes quadratic terms that account for changes in cost efficiency. Understanding this function is crucial for calculating total and average costs at different production levels.
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Properties of Functions

Quadratic Functions

Quadratic functions are polynomial functions of degree two, typically in the form ax² + bx + c. In economics, they can model cost functions where the cost changes non-linearly with production levels. The term −0.1x² in the cost function indicates diminishing returns or increasing inefficiencies as production increases, affecting the average cost calculation.
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Introduction to Polynomial Functions
Related Practice
Textbook Question

Economics


Marginal revenue

Suppose that the revenue from selling x washing machines is


r(x) = 20000(1 − 1/x) dollars.


b. Use the function r'(x) to estimate the increase in revenue that will result from increasing production from 100 machines a week to 101 machines a week.

Textbook Question

Economics


Marginal revenue

Suppose that the revenue from selling x washing machines is


r(x) = 20000(1 − 1/x) dollars.


c. Find the limit of r'(x) as x → ∞. How would you interpret this number?

Textbook Question

Economics


Marginal cost Suppose that the dollar cost of producing x washing machines is c(x) = 2000 + 100x − 0.1x².


c. Show that the marginal cost when 100 washing machines are produced is approximately the cost of producing one more washing machine after the first 100 have been made, by calculating the latter cost directly.

Textbook Question

Understanding Motion from Graphs


The accompanying figure shows the velocity v = f(t) of a particle moving on a horizontal coordinate line.


c. When does the particle move at its greatest speed?

Textbook Question

Understanding Motion from Graphs


The accompanying figure shows the velocity v = f(t) of a particle moving on a horizontal coordinate line.


d. When does the particle stand still for more than an instant?

Textbook Question

In Exercises 19–22, find the values of the derivatives.


dr/dθ |θ₌₀ if r = 2/√(4 – θ)